Degenerate nonlinear parabolic-hyperbolic equations and ... - SMAI
Degenerate nonlinear parabolic-hyperbolic equations and ... - SMAI
Degenerate nonlinear parabolic-hyperbolic equations and ... - SMAI
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<strong>Degenerate</strong> Parabolic Problems & FV Discretization Theoretical foundations Meshes, operators <strong>and</strong> scheme Discrete calculus & Convergence analysis<br />
Theoretical framework for <strong>parabolic</strong>-<strong>hyperbolic</strong> problems...<br />
Estimates easy to get (at least, formally) for approximate solutions:<br />
(existence) a priori bound on wh = A(vh) in L p (0, T; W 1,p<br />
0 (Ω)) (energy<br />
estimate) <strong>and</strong> weak compactness in L p for ∇wh = ∇A(vh)<br />
(existence) consequently, “strong compactness in space” for wh = A(vh)<br />
(Fréchet-Kolmogorov theorem)<br />
(existence) with the help of the evolution equation, “strong compactness<br />
in time” for uh = b(vh) (Fréchet-Kolmogorov)<br />
(uniqueness) very formally, given two solutions v,ˆv,<br />
multiply Eq(v)− Eq(ˆv) by sign + (v−ˆv); get <br />
Ω (b(v)−b(ˆv))+ (t) 0.<br />
(existence) Consequently, a priori L ∞ bound on uh = b(vh)<br />
(by comparison with constant solutions)<br />
Difficulties <strong>and</strong> hints to resolve them :<br />
(existence ?) No classical solutions =⇒ weak formulation<br />
(uniqueness ?) Non-uniqueness of weak solutions =⇒ selection by<br />
entropy inequalities (thus, entropy weak formulation )<br />
(uniqueness ?) Justify the formal calculation with “sign + (v−ˆv)” test<br />
function =⇒ doubling of variables following Kruzhkov (div F(v)) <strong>and</strong><br />
Carrillo (diva0(∇w))
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